The role of diffeomorphisms

2.14 The role of diffeomorphisms

As in other discrete formulations, the spatial diffeomorphism group cannot be realized exactly on the lattice, and the only obvious symmetries of a cubic lattice are discrete rotations and overall translations. The commutator computation described above indicates that one may be able recover the diffeomorphism invariance in a suitable continuum limit. Corichi and Zapata [83 ] have suggested the presence of a residual diffeomorphism symmetry in the lattice theory, under which, for example, all non-intersecting Wilson loop lattice states would be identified.

One can try to interpret the lattice theory as a manifestly diffeomorphism-invariant construction, with the lattice representing an entire diffeomorphism equivalence class of lattices embedded in the continuum [146 ]. In order to make this interpretation consistent, one should modify the functional form of either the Hamiltonian or the measure, in such a way that the commutator of two lattice Hamiltonians vanishes, as $a → 0$ .